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Design of Clutch-Slip Controller for Automatic Transmission Using Backstepping Bingzhao Z. Gao, Hong Chen, Member, IEEE, Kazushi Sanada, and Yunfeng Hu
Abstract—To improve the shift quality of vehicles with clutch-toclutch gearshifts, a nonlinear controller using backstepping technique is designed for the clutch-slip control. Model uncertainties including steady-state errors and unmodeled dynamics are also considered as additive disturbance inputs, and the controller is designed such that the error dynamics is input-to-state stable. Lookup tables, which are widely used to represent complex nonlinear characteristics of engine systems, appear in their original form in the designed nonlinear controller. Finally, the designed controller is tested on an AMESim power train simulation model. Comparisons with an existing linear algorithm are given as well. Index Terms—Automatic transmission (AT), backstepping, clutch-slip control, input-to-state stability, nonlinear control.
I. INTRODUCTION O IMPROVE fuel economy, reduce emission, and enhance driving performance, many new technologies have been introduced in the transmission area in recent years [2], such as dual clutch transmission (DCT) [3], [4] and new automatic transmissions (ATs) [5] controlling the clutches independently. In both transmissions, the change of speed ratio can be regarded as a process of one clutch to be engaged, while another being disengaged, namely, clutch-to-clutch shifts [6]. During the shift inertia phase [7], the applying (oncoming) clutch slips and the rotational speeds change intensively. The clutch-slip control during inertia phase effects shift quality (smoothness and efficiency) greatly. The clutch-slip control of stepped ratio transmission has been discussed intensively by many researches [8]. Because the clutch
T
Manuscript received July 3, 2009; revised November 20, 2009 and February 15, 2010; accepted February 22, 2010. Date of publication April 5, 2010; date of current version May 6, 2011. Recommended by Technical Editor C. R. Koch. This work was supported in part by the National Science Fund of China for Distinguished Young Scholars under Grant 60725311 and in part by the National Science and Technology Major Project under Grant 2009ZX01038-002-001. This paper was presented in part at the IEEE Conference on Decision and Control, Shanghai, China, December 16–18, 2009. B.-Z. Gao is with the State Key Laboratory of Automobile Dynamic Simulation, Jilin University, Changchun 130025, China, and also with the Department of Mechanical Engineering, Yokohama National University, Yokohama 2408501, Japan (e-mail:
[email protected]). H. Chen is with the State Key Laboratory of Automobile Dynamic Simulation, Department of Control Science and Engineering, Jilin University, Changchun 130025, China (e-mail:
[email protected]). K. Sanada is with the Department of Mechanical Engineering, Yokohama National University, Yokohama 240-8501, Japan (e-mail:
[email protected]). Y. Hu is with the Department of Control Science and Engineering, Jilin University, Changchun 130025, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2010.2045391
engagement is expected to satisfy different and sometimes conflicting objectives, such as minimizing clutch lockup time, minimizing the friction losses during the slipping phase, and ensuring a smooth acceleration of the vehicle, optimization-based algorithm is a potential solution for this problem. For example, hybrid model predictive control (HMPC) [9] and linearquadratic-based optimal control [10]–[13] are used to control the engagement of a dry clutch. Model predictive control is used in [9] so that the constrains on the control and state variables can be considered. In [12], it is pointed out that, to overcome the problem of online demanding computationally, the time evolution computed offline under different driving conditions can be fitted by polynomials for online application. Dynamic-programmingbased optimal control [14] and sequential quadratic programming (SQP) [15] are also used for gearshift operations in ATs. Although optimization-based algorithms are suitable for clutchslip control, they are always used for standing startup scenario (except for [14] and [15]), where the duration time is relatively long and there are large amount of friction losses. Different from optimal algorithms that use penalty functions to formulate multiple control objectives simultaneously, there is another kind of controller design method for clutch-slip control, in which the only control objective is to make clutch speed track the reference trajectory. In particular, for gearshift operation during driving, where the duration is much shorter than that of the startup scenario, the speed-tracking control method is widely used [6], [16]–[19]. Toward the highly nonlinear power train dynamics, such as the characteristics of engine and torque converter, and the uncertainties, such as the parameter variation of hydraulic systems and the perturbations of road-resistance torque, sliding-mode control [6], [16], μ synthesis [18], and 2-DOF control [18], [19] are used to ensure consistent control performance. Although during the shift operations speedtracking control does not consider the multiple control objectives directly, the required shift time and shift comfort can be reached by selecting proper reference trajectory; the friction losses can also be reduced by suitable engine-torque coordination during shift process [19]. This paper, therefore, will focus on the later method, i.e., to carry out clutch-speed-tracking control during the shift inertia phase of a stepped ratio AT while taking into account the system nonlinearities and uncertainties. On the other hand, backstepping has been widely used for nonlinear control-system design [20]–[24]. Furthermore, inputto-state stability (ISS) [20], [25], [26] gives the Lyapunov stability analysis of nonautonomous systems, thus the concept of ISS can be used to find a state-feedback control law that makes the closed-loop system input-to-state stable with respect to external
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TABLE I PARAMETERS FOR CONTROLLER DESIGN
Δω˙ = (C13 − C23 )μ(Δω) RN A pcb + f (Δω, ωt , ωe ) p˙cb = −
1 Kcv pcb + ib τcv τcv
(1a) (1b)
with Fig. 1.
Schematic graph of AT.
f (Δω, ωt , ωe ) = (C11 − C21 ) Tt (ωt , ωe ) + (C14 − C24 ) disturbances. Then, this paper uses backstepping to carry out the clutch-slip control problem. Model uncertainties are considered as additive disturbance inputs, and the controller is designed such that the dynamics of the speed-tracking error is input-tostate stable. The rest of the paper is organized as follows. In Section II, a dynamic model of the considered driveline is derived. The clutch-slip controller is designed based on backstepping and ISS in Section III. Based on the theoretic analysis, a systematic procedure to design the controller is then given, followed by a design example. In Section IV, the proposed controller is tested on a complete power train simulation model, and the comparison with an existing linear algorithm [19] is given as well.
II. CLUTCH SYSTEM MODELING AND PROBLEM STATEMENT We consider the power train in passenger vehicles with a twospeed AT, as schematically shown in Fig. 1. A planetary gear set is adopted as the shift gear. Two clutches are used as the actuators. Two proportional pressure valves are used to control the two clutches, respectively. When clutch A is engaged and clutch B disengaged, the power train operates on the first gear and the speed ratio is given out by i1 = 1 + (1/γ), where γ is the ratio of the teeth number of the sun gear to that of the ring gear. While clutch A is disengaged and clutch B engaged, the vehicle is driven on the second gear with a speed ratio of i2 = 1. The power-on first to second up-shift is considered here as an example. The gearshift process is divided into the torque phase where the turbine torque is transferred from clutch A to clutch B and the inertia phase where clutch B is synchronized [7]. In this paper, focus is put on the clutch-slip control during shift inertia phase. During the inertia phase, the pressure of clutch A has already been reduced to a very low level, and the dynamics of the clutch system can be described by the following equation:
Tv e (Δω, ωt ) − (C13 − C23 ) μ(Δω)RN Fs
(2)
where Δω is the speed difference of clutch B, i.e., the speed difference between the sun gear and the ring gear, pcb is the pressure of cylinder B, ib is the current of valve B, Tt is the turbine torque, Tv e is the equivalent-resistant torque delivered from the tire to the drive shaft, Cij is the constant coefficients determined by the inertia moments of vehicle and transmission shafts, and Fs denotes the return spring force of clutch B. Other parameters are defined in Table I and Fig. 3 (refer to [27] for the detailed modeling process). Due to the extreme complexity of the torque converter and the aerodynamic drag, the nonlinear function in (2) is, in general, given as lookup table (or map), which is obtained by a series of steady-state experimentations and contains inherently errors. Other modeling uncertainties include uncertain parameters, such as the vehicle mass, the road grade, the damping coefficient of shafts, etc. In this study, with the same reason in [10], the engine control (Tt ) is regarded as a noncontrolled input (it is decided by openloop algorithm based on shift timing), and the electric current of valve B ib is manipulated to make the speed difference of clutch B Δω track a reference trajectory. As mentioned earlier, the shift process should assure driving comfort and minimize the dissipated friction energy. In general, if shift duration is limited in a suitable short time, there will not be too much dissipated energy. As for the driving comfort, because the lockup of clutch tends to cause a sudden change of transmission output torque, the clutch engagement should satisfy the so-called nolurch condition [9], [10], [14], i.e., the rotational acceleration of clutch input shaft should be equal to that of the output shaft at the synchronization point. Therefore, the desired trajectory shown in Fig. 2 should satisfy the following requirements. 1) tf − t0 does not exceed the required shift time. 2) The change rate of the trajectory at tf is zero. 3) Moreover, in order to avoid control saturation, the change rate of the clutch speed at t0 should be a small value.
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where κ1 > 0 and κ2 > 0. Note that μ(x1 ) = 0 in our case and if the clutch-slip controller is well-designed, the signum of x1 does not change during one independent shift inertia phase. Hence, we can choose x2d =
−κ3 e1 − f + x˙ 1d − |b11 |κ1 e1 − |a1 μ(x1 )|κ2 e1 a1 μ(x1 )
(9)
with κ3 > 0 to guarantee Fig. 2.
|b11 | 2 |a1 μ(x1 )| 2 V˙ 1 ≤ −κ3 e21 + w + e2 . 4κ1 1 4κ2
Reference trajectory of clutch speed difference.
III. CLUTCH-SLIP CONTROLLER A. Nonlinear Controller With ISS Property In this section, we will make use of backstepping technique to derive a nonlinear controller for the problem stated earlier. The robustness of the designed controller with respect to model errors is achieved in the sense of ISS property. To do this, we rewrite the dynamics system for clutch-slip control as follows: x˙ 1 = a1 μ(x1 )x2 + f (x1 , ωe , ωt ) + b11 w1
(3a)
x˙ 2 = a2 x2 + b22 u + b21 w2
(3b)
where, x1 = Δω, x2 = pcb /1000, so that x1 and x2 are at the same order of magnitude, w1 and w2 summarize model uncertainties, and b11 and b21 are known scaling factors. Moreover a1 = (C13 − C23 )RN A × 1000
(4a)
1 a2 = − τcv
(4b)
b22
Kcv . = τcv × 1000
Hence, we conclude that the control law (9) renders the subsystem (6) ISS with respect to w1 and e2 . Since (3b) is with a linearly parameterized form, we can use e˙ 2 = a2 e2 + b22 v + b21 w2
ud =
V˙ 2 = e2 e˙ 2 ≤ a2 e22 + b22 e2 v + |b21 |κ4 e22 +
|b21 | 2 w 4κ4 2
= e2 (a2 e2 + b22 v + |b21 |κ4 e2 ) +
|b21 | 2 w 4κ4 2
(13)
V˙ 1 = e1 e˙ 1 = e1 [a1 μ(x1 )(e2 + x2d ) + f + b11 w1 − x˙ 1d ] . (7) By the use of Young’s inequality [20], the aforementioned equality becomes |b11 | 2 + w 4κ1 1
κ5 + a2 + |b21 |κ4 e2 b22
(14)
and, hence, u = ud −
κ5 + a2 + |b21 |κ4 e2 b22
(15)
then, (13) becomes |b21 | 2 w V˙ 2 ≤ −κ5 e22 + 4κ4 2
(6)
Let V1 = 1/2e21 and by differentiating it along (6), we infer
|b11 |κ1 e21
v=−
(5)
x˙ 1 = a1 μ(x1 )(e2 + x2d ) + f (x1 , ωe , ωt ) + b11 w1 .
(16)
where κ5 > 0. With the control given by (9), (12), and (15), the total closed-loop error system can be written as follows: e˙ 1 = − (κ3 + |b11 |κ1 + |a1 μ(x1 )|κ2 ) e1 + a1 μ(x1 )e2 + b11 w1
(17a)
e˙ 2 = −(κ5 + |b21 |κ4 )e2 + b21 w2
(17b)
for which we define V = V1 + V2 . Then, by exploiting (10) and (16), we have
|a1 μ(x1 )| 2 e2 4κ2
V˙ = V˙ 1 + V˙ 2
= e1 [a1 μ(x1 )x2d + f − x˙ 1d + |b11 |κ1 e1 + |a1 μ(x1 )|κ2 e1 ] |b11 | 2 |a1 μ(x1 )| 2 w + e2 4κ1 1 4κ2
(12)
where κ4 > 0. If the control law is chosen as
The control objective is to track a given smooth trajectory of x1 , which is denoted as x1d . To do this, we define the tracking error as e1 = x1 − x1d . We consider x2 as control input and determine a control law of x2d such that the tracking-error dynamics is input-to-state stable with respect to the disturbance w1 . We define e2 = x2 − x2d and rewrite (3a) as follows:
+
x˙ 2d − a2 x2d b22
for b22 (x2 , u) = 0. Because Kcv and τcv are simplified as positive constants for the controller design (see Section III-C), b22 (x2 , u) = 0 is satisfied in our case. Let V2 = (1/2)e22 and infer
(4c)
u = ib .
+ |a1 μ(x1 )|κ2 e21 +
(11)
to design the control law, where v = u − ud with
The electric current of valve B is chosen as system input
≤ e1 [a1 μ(x1 )x2d + f − x˙ 1d ] +
(10)
(8)
≤ −κ3 e21 +
|a1 μ(x1 )| − κ5 4κ2
e22 +
|b11 | 2 |b21 | 2 w + w . 4κ1 1 4κ4 2 (18)
GAO et al.: DESIGN OF CLUTCH-SLIP CONTROLLER FOR AUTOMATIC TRANSMISSION USING BACKSTEPPING
Hence, we conclude the following results for the property of the error dynamics of the designed controller. Theorem 1: Suppose that 1) κi > 0, i = 1 ∼ 5; |μ(x 1 ) − κ5 < 0. 2) |a 1 4κ 2 Then, the error dynamics of the system under controller (9), (12), and (15) is input-to-state stable, if w1 and w2 are bounded in amplitude, i.e., w1 , w2 ∈ L∞ . Proof: It follows from (18) that |a1 μ(x1 )| ˙ V ≤ − min κ3 , κ5 − e22 4κ2 |b11 | |b21 | + max , w22 (19) 4κ1 4κ4 where e = [e1 , e2 ]T and w = [w1 , e2 ]T , which shows that the error dynamics admits the input-to-state stability property [20, p. 503] if the model error w is supposed to be bounded in amplitude. Remark 3.1: It follows from (16) that |b21 | 2 V˙ 2 ≤ −2κ5 V2 + w . 4κ4 2 d 2κ 5 t |b21 | 2 2κ 5 t V2 e ≤ w e . dt 4κ4 2
(21)
Integrating it over [0, t] leads to −2κ 5 t
V2 (t) ≤ V2 (0)e
|b21 | + 4κ4
t
e−2κ 5 (t−τ ) w2 (τ )2 dτ
2 −2κ 5 t
e2 (t) ≤ e2 (0) e 2
(22)
0
and, hence, |b21 | + 2κ4
t
|b21 | w2 2∞ , 4κ4 κ5
e2 (∞) = lim s s→0
e−2κ 5 (t−τ ) w2 (τ )2 dτ.
t → ∞.
(25)
Applying the similar procedure for the tracking error e1 , we have |b11 | t −2κ 3 (t−τ ) e w1 (τ )2 dτ e1 (t)2 ≤ e1 (0)2 e−2κ 3 t + 2κ1 0 |a1 | t −2κ 3 (t−τ ) + e e2 (τ )2 |μ(x1 (τ ))|dτ (26) 2κ2 0 |b11 | w1 2∞ |a1 μm ax | e2 2∞ e1 (t) ≤ + , 4κ1 κ3 4κ2 κ3 for w1 being bounded in amplitude.
(28)
which implies e2 (∞) = 0 for w2 being an impulse signal and e2 (∞) =
b21 w ¯2 κ5 + |b21 |κ4
(29)
¯2 is the step magnitude. for w2 being a step signal, where w According to Theorem 1, Remark 3.1, and Remark 3.2, we now give the following procedure to design the clutch-slip controller in the form of (9), (12), and (15). S1) Choose tuning parameters κ3 > 0 and κ5 > 0, according to the required decay rate of the tracking error for e1 [see (23)] and e2 [see (26)], respectively; S2) Assume w2 being a step signal, determine κ4 > 0 according to (29) and the required offset of e2 . S3) For given bounds of w1 and e2 , choose κ1 > 0 and κ2 > |a1 μ(x1 )|/4κ5 such that the upper bound of the tracking offset e1 (∞) computed by (27) is acceptable. B. Implementation Issues 1) Stability Analysis Considering Pressure Observer: Until now, the clutch-slip controller was built, assuming that the clutch pressure x2 was available. Actually, however, it is not desired to use pressure sensors because of the cost and durability [27], [29]. Fortunately, a pressure observer can be constructed using the measured speed information. The discussion of the pressure estimation can be found in [27] and the designed pressure observer is exponentially stable. When the pressure observer is involved, the stability of the controller should be guaranteed under the interaction between ˆ2 , where x ˆ2 the observer and the controller. Define e3 = x2 − x denotes the estimated clutch pressure. Then, the control law (15) becomes u = ud −
and 2
b21 w2 (s) s + κ5 + |b21 |κ4
0
(23) If w2 is bounded in amplitude, i.e., w2 ∈ L∞ , then (22) becomes |b21 | w2 2∞ t −2κ 5 (t−τ ) e dτ e2 (t)2 ≤ e2 (0)2 e−2κ 5 t + 2κ4 0 (24) which implies that the tracking error e2 is bounded as e2 (t)2 ≤
Therefore, it follows from (23) and (26) that κ3 and κ5 can be chosen according to the required decay rate of the errors. And from (25) and (27), one may choose larger κ1 , κ2 , and κ4 to reduce tracking offsets. However, one should note that the larger these tuning parameters, the higher the controller gain, i.e., the choice of κi, i=1,2,4 requires the tradeoff between the tracking offset and the controller gain. Remark 3.2: We stress that (25) and (27) gives just an upper bound of the tracking offsets, if the bound of the model error is given. The real offset could be much smaller, due to the multiple use of inequalities in the aforementioned derivation. Moreover, we note that (17b) is linear time-invariant. Hence, we can compute the tracking offset for e2 by the use of the final-value theorem [28]
(20)
Upon multiplication of (20) by e2κ 5 t , it becomes
501
t→∞ (27)
= ud −
κ5 + a2 + |b21 |κ4 (ˆ x2 − x2d ) b22 κ5 + a2 + |b21 |κ4 (e2 − e3 ) b22
(30)
and consequently, the error dynamics (17) should be rewritten as follows:
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e˙ 1 = − (κ3 + |b11 |κ1 + |a1 |μ(x1 )κ2 ) e1 + a1 μ(x1 )e2 + b11 w1 (31a) e˙ 2 = −(κ5 + |b21 |κ4 )e2 + (κ5 + a2 + |b21 |κ4 )e3 + b21 w2 . (31b) If we define w2 = w2 + (κ5 + a2 + |b21 |κ4 )/b21 e3 , then we have e˙ 2 = −(κ5 + |b21 |κ4 )e2 + b21 w2
(32)
and the derivative of the Lyapunov function, i.e., (18) becomes V˙ = V˙ 1 + V˙ 2 ≤ −κ3 e21 +
|a1 |μ(x1 ) − κ5 4κ2
|b11 | 2 |b21 | 2 w + w . e22 + 4κ1 1 4κ4 2 (33)
From [27], the pressure observer is designed as follows: η=x ˆ2 − Lx1
(34a)
η˙ = (a2 − La1 μ(x1 )) (η + Lx1 ) + b22 u − Lf
(34b)
where L is the observer gain, and the estimation error is derived as follows: e˙ 3 = (a2 − La1 μ(x1 ))e3 + b21 w2 − Lb11 w1 .
(35)
It is shown that e3 is bounded when w1 and w2 are bounded [27], and consequently, w2 is also bounded. Therefore, the whole system is proved to be ISS, according to (33) and (35) . 2) Reference Trajectory: There are many methods to define the reference trajectory shown in Fig. 2, such as transfer functions or polynomials. Here, the following third-order polynomial: 2Δω0 3Δω0 (t − t0 )3 − (t − t0 )2 + Δω0 (tf − t0 )3 (tf − t0 )2 (36) is adopted to meet the requirements shown in Fig. 2, i.e., x1d (t0 ) = Δω0 , x1d (tf − t0 ) = 0 and x˙ 1d (t0 ) = x˙ 1d (tf − t0 ) = 0. Refer to Fig. 2 for the definitions of Δω0 , t0 , and tf . Then, x˙ 1d can be calculated as follows: x1d (t) =
x˙ 1d (t) =
6Δω0 6Δω0 (t − t0 )2 − (t − t0 ). 3 (tf − t0 ) (tf − t0 )2
(37)
Moreover, x˙ 2d is needed for the implementation of (12). Since μ(x1 ) and f (x1 , ωe , ωt ) are given as maps, it is impossible to obtain the explicit form of x˙ 2d by differentiating (9). Hence, we apply the input shaping technique [30], which is on occasion named as dynamic furface control (DSC) in the backstepping literature [31]. The result of (9) is labeled as x ¯2 , and passed through a first-order filter ¯2 , τ2 x˙ 2d + x2d = x
x2d (0) = x ¯2 (0)
(38)
which yields x˙ 2d =
x ¯2 − x2d . τ2
(39)
Fig. 3.
Friction characteristics of clutch plates.
3) Discussion of Different Shift Maneuvers: Until now, the power-on first to second up-shift was taken as the example to illustrate the design procedure of the clutch shift controller. Actually, the backstepping design method can be used for other clutch shift operations as well, such as the power-on down shift and power-off shift. Model (3) based on which the controller is designed satisfies the form of “strict-feedback system” [20, p. 58], i.e., ⎧ x˙ = f (x) + g(x)ξ1 ⎪ ⎪ ˙1 = f1 (x, ξ1 ) + g1 (x, ξ1 )ξ2 ⎪ ξ ⎪ ⎨ ξ˙2 = f2 (x, ξ1 , ξ2 ) + g2 (x, ξ1 , ξ2 )ξ3 (40) ⎪ .. ⎪ ⎪ . ⎪ ⎩ ξ˙k = fk (x, ξ1 , . . . , ξk ) + gk (x, ξ1 , . . . , ξk ) u where x ∈ Rn and ξ1 , . . . , ξk are scalars. In this class of systems, the nonlinearities fi and gi depend only on x, ξ1 , . . . , ξi . When the clutch slips, the delivered torque Tc is linearly parameterized with the clutch pressure pc , i.e., Tc = sign(Δω) μRN Apc .
(41)
Therefore, in the inertia phase of any shift maneuvers, the dynamic equations of the clutch system always satisfy the form of “strict-feedback system” (40), and furthermore, it can be summarized as the form of (3), which assures the application of the designed controller for other shift operations. It should be noted, however, that different shift operations may have different dynamic parameters, for example, sign(Δω) takes different values of 1 or −1. Hence, controller with different parameters is needed for different shift maneuvers. C. Controller of the Considered Clutch System Together with (37) and (38), (9), (12), and (15) constitute the proposed clutch-slip control scheme for the problem stated in Section II. Now, we present the concrete controller with all physical and tuning parameters. For simplicity, it is assumed that (τcv , Kcv ) are constant. Together with other physical parameters, they are listed in Table I. The scaling factors b11 and b21 are set to be 1. Nonlinear functions f (Δω, ωe , ωt ) and μ(Δω) are given as lookup tables in the controller. The map of μ is shown in Fig. 3, while f is given by third-order maps and examples when ωe = 200 rad/s and ωe = 500 rad/s are shown in Fig. 4. Following the procedure given in Section III-B, we first choose κ3 and κ5 to meet the requirement for the desired decay
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503
here are the estimation error of the turbine torque Tt , and the variations of road grade θg and vehicle mass m, which affect driving resistance Tv e . The estimation precision of turbine torque Tt relies on torque converter modeling [32]–[34]. Here, the estimation error of the turbine torque is assumed to be T˜t ∞ = 20 Nm, which is about ±12% of the maximum engine torque. Moreover, it is assumed that the vehicle mass is increased from 1500 kg, which is the nominal value used for controller design, to mfull = 2000 kg, which is the fully loaded mass, and the road grade angle is increased from 0◦ to 5◦ . Hence, w1 ∞ can be approximately estimated as follows: mfull g sin(θg )Rw w1 ∞ ≤ |C11 − C21 | T˜t ∞ + |C14 − C24 | Rdf = 543
Fig. 4.
Maps of f when ω e = 500 rad/s.
rate of the estimation error. It is desired that the error converges in 0.1 s, and we consider the settling time as four time constants [28], which implies 4/κ3,5 = 0.1 and results in κ3 = 40
(42)
κ5 = 40.
(43)
Then, we choose κ4 with the purpose of achieving a smaller offset of e2 . In consideration of the pressure observer, (29) that is used to calculate e2 (∞) should be rewritten as follows: e2 (∞) =
b21 w ¯ κ5 + |b21 |κ4 2
(44)
where w ¯2 is the magnitude of w2 = w2 + (κ5 + a2 + |b21 |κ4 )/ b21 e3 (see (32) for reference). It is assumed here that the model error w2 is mainly caused by the pressure estimation error e3 . According to [27], the maximum value of pressure estimation error is about 0.06 MPa. Therefore, following (44), we have e2 (∞) =
60 (κ5 + a2 + κ4 ) . κ5 + |b21 |κ4
(45)
where C11 − C21 = 25.85, C14 − C24 = 0.0911, and Rw is the tire radius. Then according to (27), and by substituting the value of e2 (∞) into e2 ∞ (since e2 is exponentially stable [according to (17b)], it is assumed that e2 (0) is a small value), e1 (∞) can be calculated as follows: |b11 | w1 2∞ |a1 μm ax | e2 2∞ + 4κ1 κ3 4κ2 κ3 1816 152 = + . κ1 κ2
e1 (∞)2 ≤
On the other hand, κ2 should satisfy, κ2 >
|a 1 |μ(x 1 ) 4κ 2
|a1 μm ax | = 0.17. 4κ5
κ4 = 10
(46)
e2 (∞) ≤ 30(×1000 Pa)
(47)
results in
(49)
− κ5 < 0, i.e., (50)
If the acceptable control offset is set to be e1 (∞) ≤ 15 rad/s
(51)
which is precise enough for this application, κ1 and κ2 can be chosen as follows: κ1 = 20
(52)
κ2 = 2
(53)
and the resulting control offset is e1 (∞) = 12.9 rad/s.
Choosing
(48)
(54)
IV. SIMULATION RESULTS A. Power Train Simulation Model
which is less than 0.06 MPa, and it is regarded as acceptable for the considered uncertainty. In order to choose the values of κ1 and κ2 , we now roughly calculate the bound of the modeling error w1 . Since power train systems admit highly nonlinear, complex dynamics and various uncertainties, it is indeed very difficult, if not impossible, to obtain a comprehensive estimation of the modeling error bound. Hence, we only consider some major uncertainties as an example to estimate the value of w1 . The major uncertainties considered
The power train simulation model is established by commercial simulation software AMESim. AMESim allows to model vehicle driveline and complete transmission, including vibrations and shift dynamics. Except for the simplified two-speed transmission, the parameters used in this paper represent a typical front-wheel-drive mid-size passenger car equipped with a 2000-cc injection gasoline engine. The constructed model captures the important transient dynamics during the vehicle shift process, such as the drive shaft oscillation and the tire slip. These dynamics are crucial for shift dynamic quality, during
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Fig. 6. Fig. 5.
Engine torque map with speed and throttle opening.
Capacity factor C (λ) and torque ratio t(λ) of the torque converter.
a tanh function of the controller design; however, they are not included in order to obtain a practically applicable controller. Moreover, time delay in control and time-varying parameters are also considered in the simulation model of the proportional valves [18], which are neglected in the controller design. 1) Engine: The work reported in this paper is primarily concerned with shift transients, and therefore, a simple engine model is used. The dynamic equation of the engine speed is represented by Ie ω˙ e + Ce ωe = Te − Tp
(55)
where Te is the engine output torque and Tp is the output torque of the converter pump. The engine output torque is simplified as a nonlinear function of the engine rotational speed ωe and the engine throttle angle θth , i.e., Te = Te (ωe , θth ). This map is shown in Fig. 5. 2) Torque Converter: When the turbine is driven forward, the dynamics of the torque converter are often characterized as Tp = C(λ)ωe2 and Tt = t(λ)Tp [32], where the capacity factor C(λ) and the torque ratio t(λ) are given in Fig. 6. 3) Planetary Gear Set: Using the submodel provided by AMESim, the planetary gear set can be conveniently modeled. The following parameters are required for the modeling setting: the inertia moment of the torque converter turbine It , the inertia moment of the ring gear Ir , the teeth number of the sun gear Zs , and the teeth number of the ring gear Zr . 4) Differential Gear Box and Drive Shaft: The gear ratio of the differential gear box is denoted as idf . The two drive shafts between the differential gear and the front wheels are represented as a torsion spring with stiffness coefficient Kl and a torsion damping with damping coefficient Cl , i.e., Ts = Kl θs + Cl θ˙s
(56)
where Ts is shaft torque and θs denotes shaft twist angle. 5) Tires: The longitudinal tire force Fx , which is usually simplified as a function of the longitudinal slip ratio Sx , rises fast when Sx increases under a threshold and declines slowly after that [35]. Here, a simple tire model of AMESim is used, and the longitudinal tire force is represented approximately as
Fx = Fxm ax tanh
2Sx dSx
.
(57)
The longitudinal slip is calculated as Sx = (Rw ωw − V )/V , where ωw is the wheel rotary velocity and V is the car body velocity. Under this operation, Fx increases until Sx approximately equals to dSx , and when Sx is greater than dSx , Fx is almost constant to be Fxm ax . In the simulations of this study, it is assumed that the vehicle is driven on the dry concrete road. 6) Road Loads: The road load consists of three parts: the grade force FG , the rolling resistant moment of tires Tw , and the aerodynamics drag FA . The resistant moment of tires Tw is regarded as constant here. The grade force can be calculated as FG = mg sin θg , where m is the vehicle mass and θg is the grade angle of the road. The aerodynamics drag is described as FA = 1/2ρ CD AA V 2 [35]. 7) Clutches and Valves: In the controller design in Section III-C, we assume the parameters (τcv , Kcv ) of the proportional pressure control valve as constant, and we also ignore the time delay of the valve. Actually, the valve has a time delay and the parameters vary according to different operating points [18]. Hence, the dynamics of the proportional valve in the power train simulation is given by ˜ cv ib (t − L ˜ cv ). τ˜cv p˙cb (t) = −pcb (t) + K
(58)
Finally, the values of the parameters used in the power train simulation are listed in Table II. Nonlinear functions ˜ cv , and K ˜ cv are given Te (ωe , θth ), C(λ), t(λ), μ(Δω), τ˜cv , L as lookup tables. B. Simulation Results The proposed clutch controller is programmed using MATLAB/Simulink and combined with the aforementioned complete power train simulation model through cosimulations. 1) Continuous Implementation: In this study, the major concern is put on the power-on first to second gear up-shift process. Fig. 7 gives the simulation results of the shift process with the nominal driving condition of Table II, i.e., the condition for the controller design. During the shift process, the engine throttle angle is adjusted to cooperate with the transmission shift. Note
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TABLE II VALUES OF SIMULATION MODEL PARAMETERS
that in real practice, when the throttle is changed, engine dynamics is much more complex than (55). In this paper, the complex engine-torque control loop is not included, and it is assumed that the engine torque is already well controlled to track the desired value. It can be seen that during the torque phase (between 7.7 and 7.94 s), the rotational speeds of the shafts do not change greatly, whereas during the inertia phase (between 7.94 and 8.34 s), the rotational speeds change intensively because of the clutch slip. During the torque phase, we assume that the timing of releasing and applying of clutches has been well guaranteed, and this part of work is omitted here. In the inertia phase, clutch B is controlled by the designed controller. The parameters of the desired trajectory are Δω0 = 420 rad/s and (tf − t0 ) = 0.4 s (see Fig. 2). After the inertia phase, when the clutch speed difference is small enough, the valve current is increased by a predetermined pattern to make clutch lockup reliably. In order to examine the shift shock, the transmission output torque and vehicle acceleration are given as well. The tracking error of the clutch slip speed during the inertia phase is plotted in Fig. 7(b). The maximum error is 7.5 rad/s, which is small enough for the vehicle clutch system. It should be pointed out that the shift process operates in the nominal driving condition, but the stiffness of the drive shaft and a tire model considering longitudinal slip are considered in the simulation model, while these are ignored in the model for designing the controller. Moreover, the time delay in control and time-varying parameters are also considered in the simulation model of the proportional valve. Then, the proposed controller is tested under the driving conditions that deviate from the nominal driving setting. The result are shown in Figs. 8 and 9, where the driving condition settings are as follows. Fig. 8 shows that the torque characteristics of the engine is enlarged by 15%, and subsequently, the capacity of the torque converter is also enlarged; the vehicle mass is increased
Fig. 7. Simulation results of the nominal driving condition (torque characteristics of engine and torque converter: standard, m: 1500 kg; θg : 0◦ , and It = 0.06 kg·m2 .)
from 1500 to 2000 kg, and the road grade angle is changed from 0◦ to 5◦ ; and the inertia of the turbine shaft is changed from 0.06 to 0.1 kg·m2 . Fig. 9 shows that the torque characteristics of the engine is reduced by 15%, and subsequently, the capacity of the torque converter is also reduced; the vehicle mass is reduced from 1500 to 1200 kg, and the road grade angle is changed from 0◦ to 5◦ ; and the inertia of the turbine shaft is changed from 0.06 to 0.03 kg·m2 . Note that although the driving conditions are changed, because the shift maneuvers are all power-on upshift when the engine load is 90%, the engine throttle control pattern of Figs. 8 and 9 are same as Fig. 7. It also should be
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Fig. 8. Simulation results of different driving condition1 (torque characteristics of engine and torque converter: standard × 115%, m: 2000 kg, θg : 5◦ , and It = 0.1 kg·m2 ).
Fig. 9. Simulation results of different driving condition2 (torque characteristics of engine and torque converter: standard × 85%, m: 1200 kg, θg : 5◦ , and It = 0.03 kg·m2 ).
noted that this paper focuses on the shift transient control and the determination of optimal shift point is not considered. As a comparison, the results of the 2-DOF linear controller in [19] are given as dashed lines of Figs. 8(b) and 9(b). In order to give a somehow fair comparison, the solid lines in Figs. 8(b) and 9(b) show the result of the proposed controller, where the same method as [19] (a third filter) is adopted for the reference trajectory generation. Although there exists a large extent of modeling errors, the maximum tracking error of the proposed controller is about 12 rad/s, which is still considered acceptable. Moreover, the comparison with the linear controller verifies
the potential benefits of the proposed nonlinear controller in achieving less tracking errors. 2) Discrete Implementation: In order to get an in-vehicle assessment of the proposed clutch-slip control system, the designed controller is discretized by a sampling rate of 100 Hz [34] with zero-order hold discretization. Furthermore, the discrete speed-sensor models [36], [37] are used to give the clutch speed ωc and the wheel speed ωw . The speed sensors are assumed to have 48 teeth and the time interval corresponding to three teeth is recorded to calculate the speeds. A relative tolerance of teeth location of 0.169% [36] and a trigger (to convert the
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The discrete implementation results are shown in Fig. 10, where the driving condition is the same as that of Fig. 8. The sampling rate causes some small-magnitude chattering of the responses and a relatively large tracking error at the beginning of the inertia phase. The tracking errors, however, decay to the bound of ±12 rad/s rapidly. It can also be seen that the fluctuation magnitude of transmission output torque is less than 50 Nm, which is considered acceptable for a fully loaded midsize car shifting on a slope. V. CONCLUSION Backstepping technique has been used for clutch-slip control during shift inertia phase of a stepped ratio AT. The complex nonlinear characteristics of engine systems appear in their original form of lookup tables. Model uncertainties including steady-state errors and unmodeled dynamics are considered as additive disturbance inputs and the controller has been designed such that the error dynamics is input-to-state stable. The designed controller has been tested on an AMESim power train simulation model, which contains complete drive train and clutch actuators. Comparison results with the existing linear controller verify the potential benefits of the proposed nonlinear controller in achieving less tracking error. It has also been demonstrated that the controller is robust to driving condition variations, such as change of vehicle mass and road grade. Moreover, the discrete implementations show the potential of the proposed controller to be used in real vehicles. REFERENCES
Fig. 10. Simulation results of discrete implementation (torque characteristics of engine and torque converter: standard × 115%, m: 2000 kg, road slope: 5◦ , It = 0.1 kg·m2 ).
analog signal into the square-wave signal) randomness of 1.5% are considered. Note that in the case of discrete implementation, which is a sampled-data system, there is some relatively large chatters because of the system discretization. The gains have to be adjusted to restrain the chatters, and the tuned results are κ1 = 12, κ4 = 7,
κ2 = 1.2,
κ3 = 24
κ5 = 28.
(59)
The time constant τ2 of filter (39) is chosen as τ2 = 0.05 s.
(60)
[1] B.-Z. Gao, H. Chen, Y. Ma, and K. Sanada, “Clutch slip control of automatic transmission using nonlinear method,” in Proc. 48th IEEE Conf. Decis. Control, Shanghai, China, 2009, pp. 7651–7656. [2] Z. Sun and K. Hebbale, “Challenges and opportunities in automotive transmission control,” in Proc. Amer. Control Conf., 2005, vol. 5, pp. 3284– 3289. [3] B. Matthes and F. Guenter, “ Dual clutch transmissions—Lessons learned and future potential,” SAE Technical Paper 2005-01-1021, 2005. [4] M. Kulkarni, T. Shim, and Y. Zhang, “Shift dynamics and control of dualclutch transmissions,” Mech. Mach. Theory, vol. 42, no. 2, pp. 168–182, 2007. [5] T. Minowa, T. Ochi, H. Kuroiwa, and K. Liu, “ Smooth gear shift control technology for clutch-to-clutch shifting,” SAE Technical Paper 1999-011054, 1999. [6] D. Cho, “Nonlinear control methods for automotive powertrain systems,” Ph.D. dissertation, MIT, Cambridge, MA, 1987. [7] M. Goetz, M. C. Levesley, and D. A. Crolla, “Dynamics and control of gearshifts on twin-clutch transmissions,” Proc. Inst. Mech. Eng., Part D: J. Automobile Eng., vol. 219, no. 8, pp. 951–963, 2005. [8] L. Glielmo, L. Iannelli, V. Vacca, and F. Vasca, “Gearshift control for automated manual transmissions,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 1, pp. 17–26, Feb. 2006. [9] A. Bemporad, F. Borrelli, L. Glielmo, and F. Vasca, “Hybrid control of dry clutch engagement,” presented at the Eur. Control Conf., Porto, Portugal, 2001. [10] P. Dolcini and H. B´echart, “Observer-based optimal control of dry clutch engagement,” in Proc. 44th IEEE Conf. Decis. Control, Seville, Spain, 2005, pp. 440–445. [11] P. Dolcini, C. C. Wit, and H. B´echart, “Lurch avoidance strategy and its implementation in amt vehicles,” Mechatronics, vol. 18, no. 5/6, pp. 289– 300, 2008. [12] L. Glielmo and F. Vasca, “ Optimal control of dry clutch engagement,” SAE Technical Paper 2000-01-0837, 2000. [13] F. Garofalo, L. Glielmo, L. Iannelli, and F. Vasca, “Optimal tracking for automotive dry clutch engagement,” presented at the 15th IFAC Congr., Barcelona, Spain, 2002.
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Bingzhao Z. Gao received the B.S. and M.S. degrees from Jilin University of Technology, Changchun, China, in 1998 and 2002, respectively, and the Ph.D. degree in mechanical engineering from Yokohama National University, Yokohama, Japan, and control engineering from Jilin University, in 2009. He is currently a Lecturer in the State Key Laboratory of Automobile Dynamic Simulation, Jilin University. His research interests include vehicle power train control and vehicle stability control.
Hong Chen (M’02) received the B.S. and M.S. degrees in process control from Zhejiang University, Hangzhou, China, in 1983 and 1986, respectively, and the Ph.D. degree from the University of Stuttgart, Stuttgart, Germany, in 1997. In 1986, she joined Jilin University of Technology, Changchun, China, where she became an Associate Professor in 1998 and has been a Professor since 1999. In 1992, she was a Visiting Scholar at the University of Karlsruhe, Karlsruhe, Germany. From 1993 to 1997, she was a ‘Wissenschaftlicher Mitarbeiter” at the Institut f¨ur Systemdynamik und Regelungstechnik, University of Stuttgart, where she was a Visiting Professor (German Research Foundation (DFG) Scholarship) at the Institute for Systems Theory in Engineering in spring 2001. Her current research interests include model predictive control, optimal and robust control, and applications in process engineering and mechatronic systems.
Kazushi Sanada received the B.S. and M.S. degrees in control engineering and the Ph.D. degree in mechanical engineering from Tokyo Institute of Technology, Tokyo, Japan, in 1981, 1986, and 1996, respectively. He was a Research Assistant in the Department of Control Engineering, Tokyo Institute of Technology. In 1998, he joined the Department of Mechanical Engineering and Materials Science, Yokohama National University, Yokohama, Japan, as an Associate Professor, where he has been a Professor since 2004. His current research interests include modeling and control of mechanical systems, robust control, and fluid power systems.
Yunfeng Hu received the B.S. degree in applied mathematics from Changchun Normal University, Changchun, China, in 2006, and the M.S. degree in basic mathematics from Jilin University, Changchun, in 2008, where he is currently working toward the Ph.D. degree in the Department of Control Science and Engineering. His current research interests include engine control.
